Dirichlet series and hyperelliptic curves

For a fixed hyperelliptic curve C given by the equation y(2) = f(x) with f is an element of Z[x] having distinct roots and degree at least 5, we study the variation of rational points on the quadratic twists C-m whose equation is given by my(2) = f(x). More precisely, we study the Dirichlet series D-f(s) = Sigma(m not equal 0)' #C-m(Q)vertical bar m vertical bar(-s) where the summation is over all non-zero squarefree integers. We show that D-f(s) converges for R(s) > 1. We extend its range of convergence assuming the ABC conjecture. This leads us to study related Dirichlet series attached to binary forms. We are then led to investigate the variation of rational points on twists of superelliptic curves. We apply this study to certain classical problems of analytic number theory such as the number of powerfree values of a fixed polynomial in Z[x].